high-speed next generation multi-wall carbon nanotube...
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HIGH-SPEEDNEXT GENERATIONMULTI-WALL CARBON NANOTUBEINTERCONNECT SIMULATORA COLLABORATIVE PROJECT BY
DALLAS MANN (CE) – HEAD OF PROGRAMMING
MIKE BRUNGARDT (EE) – HEAD OF RESEARCH
SUPERVISING PROFESSOR: SOURAJEET ROY
PRESENTATION OUTLINE
• Motivation of CNT• Moores Law
• Structural, Thermal, and Electrical Properties
• Applications
• Breaking down a CNT simulation• Modified Nodal Analysis/Circuit Stamps
• Resistor & Capacitor example
• Solving MNA Matrices• Frequency Doman
• Time Domain
• Connection to our project• MWCNT vs SWCNT
• MCC to ESC
• Results
• Steps for next semester• Increasing the number of conductors
• Different Dimensions
WHY CARBON NANOTUBES
• Moore’s Law – The observation that the amount
of transistors on a chip will double every two
years since 1965
• What was once in the matter of thousands, is now
in the multi billions, and still increasing
exponentially
• With more a more dense concentration of
transistors using roughly the same amount of
power innovations are needed in both power
consumption, and the thermal and electrical
properties of their interconnects
• To understand why CNT’s have these properties it is
important to understand their structure
CARBON NANOTUBES ARE FASCINATING:STRENGTH, THERMAL, AND ELECTRICAL PROPERTIES
STRENGTHWHAT
• CNT’s are one of the strongest
structures known to man at around
1TPa
• For reference, steal’s strength is
around 200 Gpa
WHY
• Carbon Nanotubes are mainly
composed of Sp2 bonds, which are
stronger then diamond’s Sp3 bonds
THERMAL CONDUCTIVITY
• Thermal Conductivity– [W per m-1K-1] the transport of energy in the form of heat though a body of mass as the result of a temperature gradient.
• Copper has the thermal conductivity of 385 Wm-
1K-1
• A SWCNT has the thermal conductivity of 3500 Wm-1K-1
ELECTRICAL CONDUCTIVITY• Electrical Conductivity – [S/m] the inverse of resistance, or how easy it is
for electricity to transfer across a structure or conduct electricity
• CNT’s display different properties depending on how the sheet is rolled.
• Armchair ɸ = 0
• Treated as metallic
• Can carry an electric Current density of 4GA/cm2
• Copper’s maximum current density is only 4MA/cm2
• Limited by electromigration
• Zig-Zag ɸ = 30
• Treated as though a semiconductor
• Chiral 0 < ɸ < 30
• Treated as though a semiconductor
APPLICATIONS: TODAY AND THE FUTURE
Current Applications
• Atomic-Force Microscopes
• CNT’s have been used to make probing tips
• Anasys Instruments
• Material Sciences
• Creates a composite material composed of epoxy bonded with
CNT that is 20%-30% stronger then other composite materials
• Amroy Europe Oy
• Tissue Engineering
• CNT’s have been used to act as a scaffolding for bone growth
• Rice University and Radboud University (Netherlands)
• Water treatment
• A MineralWater System using Nanomeshn Purification Technology
has created to filter water without the use of heat, chemicals, or
power
• Seldon Technologies
Future Applications:
• Batteries with 10x the life
• Silicon coated CNT in anodes for Lithium-Ion Batteries, preventing the
expansion of Silicon based anodes
• North Carolina State University
• Artificial Muscles
• Are 200x stronger then natural muscles with the same concentration
• University of Texas - Dallas
• Cancer Detection and Treatment• Interconnection of CNTs and gold nanoparticles to create a sensor to indicate the presence of oral Cancer
• In lab tests have been proven to destroy breast cancer tumors
• University of Connecticut
• Microelectronics
• Current copper interconnects are on the order of tens to hundreds of nanometers in
diameter
• One mode of failure in modern electronics is due to electromigration effects on these
copper interconnects
BREAKING DOWN THE SIMULATION
Motivation and Aim of the Simulator
• It is important to note that a carbon nanotube can be made
with a model composed of resistors, inductors, capacitors
known as a lumped sum model
• We wanted to make a simulation similar to PI-SPICE or
Cadence minus the GUI
• Didn’t want to waste time/resources/memory/complexity on
anything we did not need for this project
Internally SPICE is a two step process:
• First – Formulating all circuit elements taken in from a netlist and put
into a system of mathematical equations
• Depends on:
• Type of circuit Element
• Node in
• Node out
• Value of Circuit Element
• Second – Creating a solution for these equations
• Found using
• Initial Voltages
• Initial Currents
EXPLANATION BEHIND MODIFIED NODAL ANALYSIS
The laws of KCL and KVL produce:
• Basic Circuit elements in the Time domain:
• Resistors: v=Ri=i/g
• Capacitors: I = C dv/dt
• Inductors: v=L di/dt
• Basic Circuit elements in the Frequency
domain:
• Resistors: V=Ri=I/g
• Capacitors: I = sCV-CV0
• Inductors: V=sLI-LI0
• Nodal Formulation, made from Nodal
Analysis (KCL) can be easy to model
• Current going in is the same as going out
• This is the Creation of Modified Nodal
analysis, MNA for short
• Modified nodal Analysis can take any circuit
element (Resistors, inductors, capacitors, op-
amps, MOFSET, transistors, etc.)
• Turns each into a series of unique algebraic
elements
• This is called a stamp
HOW MNA MAKES STAMPS: RESISTORS
• Using KCL and Ohm’s Law we can write:
{G(Va – Vb) + …… = 0 X = [Va] ath Row
{G(Vb – Va) + …… = 0 [Vb] bth Row
• MNA Creates the following Matrices for any general
circuit:
[G][X] + [C] d[X]/dt = [B]
• [ G, -G ] [Va] satisfies G(Va - Vb) = 0
• [ -G, G ] [Vb] satisfies G(Vb - Va) = 0
• G, C, B, are the matrices where we insert each
components stamp
• G – non-derivative terms
• C – derivative terms
• B – Set DC Voltages
HOW MNA MAKES STAMPS: CAPACITORS
• Using KCL and Ohm’s Law we can write:
{C(d(Va - Vb))/dt + …… = 0 X = [Va] ath Row
{C(d(Vb - Va))/dt + …… = 0 [Vb] bth Row
• [ C, -C ] [d(Va/dt)] satisfies C(d(Va - Vb))/dt = 0
• [ -C, C ] [d(Vb/dt)] satisfies C(d(Vb - Va))/dt = 0
SOLVING IN THE FREQUENCY DOMAIN: INTRO
• [G][X] + [C][dx/dt] = B(t) Time Domain
• As we have seen sometimes Freq needed
• Get there by taking a Laplace transform
• GX(s)+(xS(s)-x0) = B(s) (G+sC)X(s) = B(s)
• Which turns to the linear system of equations:
AX = B
• One common approach to solve this is Gaussian
Elimination
• It is important to note that A
represents (G+sC) making
A-1 difficult for both us and
the computer
• A is frequency dependent
• 1000 frequency points would
take 1000 inversions of x
LU DECOMPOSITION
• LU Decomposition breaks up the A
matrix into 2 parts L and U
• L – Lower Triangular
• U – Upper Triangular
• O(n3)
• Todays Fastest algorithm is
between O(n3)O(n2.2)
FORWARDS - BACKWARDS SUBSTITUTION
• What do we do with the L and U
matrices?
• Ax=B can be written as LUx=B
• We can treat Ux as its own matrix y
• Our Ax=B is now Ly=B
• Forwards Substitution
• O(n2/2)
• With y known we can now solve Ux=Y
• Backwards Substitution
• O(n2/2)
TIME DOMAIN
• When working in the time domain we use linear
multistep methods to solve [G][X]+[C][dX/dt] = [B]
• Forward Euler (Explicit)
• dx(tk)/dt = (x(tk+1)-x(tk))/(tk+1-tk)
• Backward Euler (Implicit)
• dx(tk)/dt = (x(tk)-x(tk-1))/(tk-tk-1)
• Trapezoidal Rule (Implicit)
• 1/2 [dx(tk)/dt + dx(tk-1)/dt) = (x(tk)-x(tk-1))/(tk-tk-1)
• We only want to use Implicit solving methods
• Trapezoidal rule is guaranteed to be stable
• Implicit methods are more computationally expensive
• Increased computational complexity is a trade off for
increased accuracy
SOLVING IN THE TIME DOMAIN: BACKWARD EULER
• GX(tk) + C dX(tk)/dt = B(tk)
• GX(tk) + C (X(tk)-X(tk-1))/h) = B(tk)
• [G+C/h]X(tk) = B(tk)+C/hX(tk-1)
• Simple(r) version of other numerical integration techniques
• Less accurate then other implicit solving techniques
• Error propagates from the first time point to the last
SOLVING IN THE TIME DOMAIN: TRAPEZOIDAL
• GX(tk) + C dX(tk)/dt = B(tk)
• GX(tk-1) + C dX(tk-1)/dt = B(tk-1)
• (G/2)(X(tk)+X(tK-1)) + C X(tk)-X(tk-1)/n = (B(tk)-B(tk-1))/2
• [G/2 + C/h]X(tk) = (B(tk)+B(tk+1))/2 + (C/h – G/2)X(tk-1)
• More accurate than backward Euler
• More computationally complex
• 2nd order numerical integration technique
MULTIWALLED CARBON NANOTUBE (MWCNT)
• It is important to know that for our
project we are using a MULTI-WALL
carbon nanotube (MWCNT)
• Single Walled Carbon Nanotubes
SWCNT have an array of problems
• Random Chirality
• Brings unwanted semi conductive
properties
• Intrinsic Resistance of 6.5KΩ
• Hard to grow dense amount of bundles
• MWCNT’s can be described as many
concentric shells of cap-less SWCNT
• In our model we use up to 30 shells
BASING OUR RESEARCH
Part 1: Conducting Channels & RLC
• Modeling and Fast Simulation of
Multiwalled Carbon Nanotube Interconnects
• Min Tang, Member, IEEE, and Junfa Mao,
Fellow, IEEE
Part 2: Refining Previous paper + G & σ
• Circuit Modeling and Performance Analysis of
Multi-Walled Carbon Nanotube Interconnects
• Hong Li, Student Member, IEEE, Wen-Yan Yin,
Senior Member, IEEE, Kaustav Banerjee, Senior
Member, IEEE, and Jun-Fa Mao, Senior Member,
IEEE
MODELING MWCNT: EVERY RLC TO ACCOUNT FOR
• As shown a previous slide, a CNT can be represented with
a combination of Resistors, Capacitors and Inductors.
• Each shell has its own RLC
• There are two outside resistances just by connecting the
MWCNT to the probes
• Imperfect contact Resistance (RMC)
• When connecting a MWCNT metal must be deposited to
connect to all the shells
• Quantum Contact Resistance (RQ)
• Intrinsic resistance due to the fabrication process
• Because this deals shells near atomic levels the electrons
make “jumps from one point to another”
• Inside Each MWCNT section there are individual
Resistances, Capacitances, and inductances
• Resistances:
• Scattering Resistance (Rs)
• When connecting the anode to shells, electrons go from
following a single path to scattering
• Inductances
• Kinetic Inductance (LK)
• The inertia of the electron
• Fixed at 8nH/um 40pH/Section
• Magnetic Inductance (LM)
• Magnetic field created by moving electrons
• Incredibly small (2pH/um 100fH/section)
• Capacitances
• Quantum Capacitance (CQ)
• Capacitance between shells effected by the electron density
• Electrostatic Capacitance between shells (CS)
• Capacitance between shells
• Electrostatic Capacitance from the ground plane (CE)
• Capacitance between the dielectric and the MWCNT
MODELING MWCNT: MCC AND ESC MODELSMCC Model – Multi Conductor Circuit (Step 1)
• Shows every circuit element from each
section of each shell
• Pros
• Used for performance prediction
• Accounts for inter-shell conductivity
• Cons
• Slower process, harder to model
• Time and memory issues
ESC Model – Equivalent Single Conductor (Step 2)
• Simplifies each shell into a single one line circuit
• Pros
• Much faster then MCC model
• Simpler then the ESC model, O(n3/3) smaller
• Cons
• Approximations have to be made, thus the circuit is
less accurate
• Does not account for inter-shell conductivity
CONVERTING MCC TO ESCTo convert an MCC to an ESC
• RMC, RQ, RS combine in parallel
• Lk adds in parallel
• Again LM is neglected
• CQ and Cs follow a recursive formula
• There is only one CE and is kept as is
Per Unit Length (p.u.l.)
• It is important to be aware of the units
or these RLC’s
• Often we had numbers seemingly way
out of range
• Numbers were often Ω/m, F/m, H/m
• Need to be Ω/um, F/um, H/um
• OR even better, with 200 sections p.u.l.
• Ω/Sec, F/um, H,um
TIME DELAYSIMULATION: THEORY VS. CADENCE VS. OURS
THEORETICAL:
FROM RESEARCH PAPEROUR CIRCUIT SIMULATOR &
CADENCE MODEL:
WHERE WE’RE AT AND WHERE WE WANT TO BECurrently:
• We have produced a simulator that is able to:
• Solve Complex RLC circuits with transistors in time
and frequency domain
• Simulates a MWCNT with 30 shells using the ESC
model
• Simulation runs on any modern laptop
• Currently runs a simulation for a single
conductor
• Have not used any of our allotted Budget
Next Semester
• Solve MCC model
• Add in the dimension of dielectric conductance
as well as inter-shell conductance
• Run our simulation on a supercomputer for a
accurate, time-efficient, full MCC model
• We wish to run our simulation on 1,2,3…n
multiple conductors
• Will have to allot funds from budget for
supercomputer use
QUESTIONS?